d-collapsibility is NP-complete for d greater or equal to 4

TL;DR

This paper proves that determining d-collapsibility of simplicial complexes is NP-complete for d ≥ 4, highlighting computational complexity differences and exploring algorithmic recognition for various d values.

Contribution

It establishes NP-completeness of d-collapsibility for d ≥ 4 and analyzes the effectiveness of greedy algorithms for recognizing d-collapsibility.

Findings

NP-complete for d ≥ 4

Polynomial-time for d ≤ 2

Greedy algorithm works for d ≤ 2 but not for d ≥ 3

Abstract

A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing (collapsing) a face of dimension at most d-1 that is contained in a unique maximal face. We prove that the algorithmic question whether a given simplicial complex is d-collapsible is NP-complete for d greater or equal to 4 and polynomial time solvable for d at most 2. As an intermediate step, we prove that d-collapsibility can be recognized by the greedy algorithm for d at most 2, but the greedy algorithm does not work for d greater or equal 3. A simplicial complex is d-representable if it is the nerve of a collection of convex sets in R^d. The main motivation for studying d-collapsible complexes is that every d-representable complex is d-collapsible. We also observe that known results imply that analogical algorithmic question for d-representable complexes is NP-hard for d greater or equal to 2.